Paths, Ends and The Separation Problem for Infinite Graphs

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that this problem is decidable for every highly computable graph with finitely many ends. Using this result, we demonstrate that König’s Infinity Lemma is effective for such graphs. We also apply it to analyze the complexity of the Eulerian Path Problem for infinite graphs, showing that much of its complexity arises from counting ends. Indeed, the Eulerian Path Problem becomes strictly easier when restricted to graphs with a fixed number of ends. Under this restriction, we provide a complete characterization of the problem. Finally, we study the Separation Problem in a uniform setting (i.e., where the graph is also part of the input) and offer a nearly complete characterization of its complexity and its relationship to counting the number of ends.

💡 Research Summary

The paper introduces and studies the “Separation Problem” for infinite, locally finite, connected graphs. Given a finite set of edges E in a graph G, the problem asks whether the removal of E disconnects G into at least two infinite connected components. This notion is tightly linked to the classical concept of “ends” of a graph, defined as the supremum of the number of infinite components that can arise after deleting any finite edge set.

The authors work primarily with highly computable graphs—graphs whose vertex set, adjacency relation, and degree function are all computable. For a fixed highly computable graph G they define three central objects: (i) Comp_G(E), the number of infinite components of G \ E; (ii) Sep(G), the collection of finite edge sets that separate G into at least two infinite components; and (iii) Sepmax(G), the collection of edge sets that achieve the maximal possible number of infinite components, i.e., the number of ends of G.

The first technical contribution is an upper‑semi‑computable procedure for Comp_G(E). By exploring neighborhoods of E up to distance n, one can compute a non‑increasing sequence Comp_G,n(E) that stabilizes at Comp_G(E). Consequently, membership in Sep(G) is a Π⁰₁ condition (∀n Comp_G,n(E) ≥ 2). The authors prove this bound is tight: there exists a highly computable graph with infinitely many ends for which Sep(G) is Π⁰₁‑complete.

When the number of ends is finite, the situation improves dramatically. The authors show that Comp_G(E) becomes fully computable, making both Sep(G) and Sepmax(G) decidable. Moreover, Sepmax(G) is uniformly equivalent (via m‑reductions) to the pair (Sep(G), Ends(G)). Thus, knowing Sepmax(G) yields both the set of separating edge sets and the exact number of ends, and conversely, Sep(G) together with Ends(G) determines Sepmax(G).

Two major applications are explored. First, the infinite‑path existence problem: for highly computable graphs with finitely many ends, the graph is infinite if and only if it contains a computable infinite simple path. Moreover, the extension problem—deciding whether a given finite path can be extended to an infinite simple path—is decidable under the same hypothesis. This provides an effective version of König’s Infinity Lemma for such graphs.

Second, the Eulerian Path Problem for infinite graphs. The authors observe that a graph with more than two ends cannot have an Eulerian path, so the analysis splits into three cases: one‑ended graphs, two‑ended graphs, and the uniform setting where the graph is part of the input. For highly computable one‑ended graphs, deciding the existence of a one‑way Eulerian path is d.c.e.‑complete (difference of two c.e. sets), while deciding a two‑way Eulerian path is Π⁰₁‑complete. For highly computable two‑ended graphs, the problem attains exactly the m‑degrees of all Δ⁰₂ sets, i.e., it is as hard as any Δ⁰₂ problem under many‑one reductions. The paper also notes that for automatic one‑ended graphs the problem becomes decidable, illustrating the impact of stronger graph presentations.

The uniform version of the Separation Problem (where the graph description is part of the input) is analyzed in depth. The authors give a near‑complete complexity classification, showing that computing Sep(G) is Π⁰₁‑complete in general, while Sepmax(G) is Π⁰₂‑complete when the number of ends is unbounded. They also relate these complexities to the classical arithmetic hierarchy sets Inf (infinite c.e. sets) and Fin (finite c.e. sets).

In the final sections the paper connects its results to earlier work on infinite Hamiltonian paths in cubes and on Eulerian infinite paths, demonstrating that the Separation Problem serves as a unifying tool for many decidability questions in infinite graph theory. The authors conclude by suggesting further research directions, such as extending the analysis to automatic graphs, exploring other structural parameters (planarity, bounded degree), and investigating the interplay between ends and other combinatorial properties.

Overall, the work provides a systematic bridge between the topological notion of ends and the algorithmic complexity of fundamental graph problems, establishing decidability and hardness results that depend precisely on the number of ends of the underlying infinite graph.